In triangle ABC, the measure of /_A is 90^(@),AB=10, and BC=16. Triangle DEF is similar to triangle ABC, where vertices D,E, and F correspond to vertices A,B, and C, respectively, and each side of triangle DEF is 2 times the length of the corresponding side of triangle ABC. What is the value of sin F ?

Question

In triangle  ABC, the measure of  /_A is  90^(@),AB=10, and  BC=16. Triangle  DEF is similar to triangle  ABC, where vertices  D,E, and  F correspond to vertices  A,B, and  C, respectively, and each side of triangle  DEF is 2 times the length of the corresponding side of triangle  ABC. What is the value of  sin F ?

Bytelearn · Accepted Answer

Identify sides of triangle ABC: Identify the sides of triangle ABC. Since triangle ABC is a right triangle with /_A being 90 degrees, we can use the Pythagorean theorem to find the length of side AC. AB^2 + AC^2 = BC^2Calculate length of side AC: Calculate the length of side AC using the Pythagorean theorem. 10^2 + AC^2 = 16^2 100 + AC^2 = 256 AC^2 = 256 - 100 AC^2 = 156 AC = sqrt(156) AC = sqrt(4 * 39) AC = 2 * sqrt(39)Determine sides of triangle DEF: Determine the sides of triangle DEF. Since triangle DEF is similar to triangle ABC and each side of triangle DEF is 2 times the length of the corresponding side of triangle ABC, we have: DE = 2 * AB = 2 * 10 = 20 EF = 2 * BC = 2 * 16 = 32 DF = 2 * AC = 2 * 2 * sqrt(39) = 4 * sqrt(39)Find sin F in triangle DEF: Find the value of sin F in triangle DEF. Since triangle DEF is similar to triangle ABC, the angles are the same. Therefore, /_F in triangle DEF corresponds to /_C in triangle ABC. In a right triangle, sin of an angle is the ratio of the opposite side to the hypotenuse. sin F = opposite side to angle F / hypotenuse of triangle DEF sin F = DE / EFCalculate value of sin F: Calculate the value of sin F. sin F = DE / EF sin F = 20 / 32 sin F = 5 / 8

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